A mathematical and numerical framework for magnetoacoustic tomography with magnetic induction

TL;DR

This paper develops a comprehensive mathematical and numerical framework for magnetoacoustic tomography with magnetic induction, enabling accurate reconstruction of tissue conductivity from acoustic and electromagnetic measurements.

Contribution

It introduces new methods for reconstructing conductivity, including an optimal control approach, a fixed point method, and a PDE-based scheme, with proven convergence and stability.

Findings

The methods accurately reconstruct conductivity from noisy data.

The schemes demonstrate stability and resolution in numerical tests.

Comparison shows the effectiveness of each reconstruction approach.

Abstract

We provide a mathematical analysis and a numerical framework for magnetoacoustic tomography with magnetic induction. The imaging problem is to reconstruct the conductivity distribution of biological tissue from measurements of the Lorentz force induced tissue vibration. We begin with reconstructing from the acoustic measurements the divergence of the Lorentz force, which is acting as the source term in the acoustic wave equation. Then we recover the electric current density from the divergence of the Lorentz force. To solve the nonlinear inverse conductivity problem, we introduce an optimal control method for reconstructing the conductivity from the electric current density. We prove its convergence and stability. We also present a point fixed approach and prove its convergence to the true solution. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the electric current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. Finally, we test the three schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance.